The BGL model is a version of the 2HDM where the scalar interactions with
fermions violate flavour – meaning, unlike the interactions of the photon
and \(Z\)  boson, the neutral scalars in the BGL model “jump families" like
the \(W\)  boson does. In the more usual versions of the 2HDM the existence of FCNCs in the scalar sector is avoided by imposing a \(\mathbb {Z}_2\)  [1]}, [2]}
or U(1) [3]} symmetry on the model (see also [4]}, [5]}), which forces each fermion of the same charge to couple to a single doublet, thereby preventing FCNCs. The reason for doing this in the first place is the fact that tree-level mediated FCNCs would make significant contributions to flavour sector
observables such as the mass differences of the \(K^0\) , \(B_d\)  or \(B_s\)  mesons,
or to the \(\epsilon _K\)  quantity, ruining an agreement found within the SM for
those quantities – unless the masses of the new scalars are all of order TeV,
or the FCNC Yukawa couplings are tuned to be very small.
